Seminars and Colloquia by Series

Representations and approximations of hyperbolicity cones

Series
Algebra Seminar
Time
Monday, August 20, 2012 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Daniel PlaumannUniversity of Konstanz
Hyperbolic polynomials are real polynomials that can be thought of as generalized determinants. Each such polynomial determines a convex cone, the hyperbolicity cone. It is an open problem whether every hyperbolicity cone can be realized as a linear slice of the cone of psd matrices. We discuss the state of the art on this problem and describe an inner approximation for a hyperbolicity cone via a sums of squares relaxation that becomes exact if the hyperbolic polynomial possesses a symmetric determinantal representation. (Based on work in progress with Cynthia Vinzant)

Symmetric Groebner bases

Series
Algebra Seminar
Time
Monday, May 21, 2012 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Chris HillarUC Berkeley
We discuss the theory of symmetric Groebner bases, a concept allowing one to prove Noetherianity results for symmetric ideals in polynomial rings with an infinite number of variables. We also explain applications of these objects to other fields such as algebraic statistics, and we discuss some methods for computing with them on a computer. Some of this is joint work with Matthias Aschenbrener and Seth Sullivant.

Overconvergent Lattices and Berkovich Spaces

Series
Algebra Seminar
Time
Tuesday, April 24, 2012 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Andrew DudzikUC Berkeley
The construction of the Berkovich space associated to a rigid analytic variety can be understood in a general topological framework as a type of local compactification or uniform completion, and more generally in terms of filters on a lattice. I will discuss this viewpoint, as well as connections to Huber's theory of adic spaces, and draw parallels with the usual metric completion of $\mathbb{Q}$.

Cellular Cuts, Flows, Critical Groups, and Cocritical Groups

Series
Algebra Seminar
Time
Tuesday, April 17, 2012 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jeremy MartinUniversity of Kansas
The critical group of a graph G is an abelian group K(G) whose order is the number of spanning forests of G. As shown by Bacher, de la Harpe and Nagnibeda, the group K(G) has several equivalent presentations in terms of the lattices of integer cuts and flows on G. The motivation for this talk is to generalize this theory from graphs to CW-complexes, building on our earlier work on cellular spanning forests. A feature of the higher-dimensional case is the breaking of symmetry between cuts and flows. Accordingly, we introduce and study two invariants of X: the critical group K(X) and the cocritical group K^*(X), As in the graph case, these are defined in terms of combinatorial Laplacian operators, but they are no longer isomorphic; rather, the relationship between them is expressed in terms of short exact sequences involving torsion homology. In the special case that X is a graph, torsion vanishes and all group invariants are isomorphic, recovering the theorem of Bacher, de la Harpe and Nagnibeda. This is joint work with Art Duval (University of Texas, El Paso) and Caroline Klivans (Brown University).

Galois groups of Schubert problems of lines are at least alternating

Series
Algebra Seminar
Time
Tuesday, April 10, 2012 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Abraham Martin del CampoTexas A&M
The Galois group of a problem in enumerative geometry is a subtle invariant that encodes special structures in the set of solutions. This invariant was first introduced by Jordan in 1870. In 1979, Harris showed that the Galois group of such problems coincides with the monodromy group of the total space. These geometric invariants are difficult to determine in general. However, a consequence of Vakil's geometric Littlewood-Richardson rule is a combinatorial criterion to determine if a Schubert problem on a Grassmannian contains at least the alternating group. Using Vakil's criterion, we showed that for Schubert problems of lines, the Galois group is at least the alternating group.

Real solving and certification

Series
Algebra Seminar
Time
Tuesday, April 3, 2012 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jonathan HauensteinTexas A&M
In many applications in engineering and physics, one is interested in computing real solutions to systems of equations. This talk will explore numerical approaches for approximating solutions to systems of polynomial and polynomial-exponential equations. We will then discuss using certification methods based on Smale's alpha-theory to rigorously determine if the corresponding solutions are real. Examples from kinematics, electrical engineering, and string theory will be used to demonstrate the ideas.

The central curve of a linear program

Series
Algebra Seminar
Time
Tuesday, February 28, 2012 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Cynthia VinzantUniversity of Michigan
The central curve of a linear program is an algebraic curve specified by a hyperplane arrangement and a cost vector. This curve is the union of the various central paths for minimizing or maximizing the cost function over any region in this hyperplane arrangement. I will discuss the algebraic properties of this curve and its beautiful global geometry, both of which are controlled by the corresponding matroid and hyperplane arrangement.

Integral Closure Presentations and Membership

Series
Algebra Seminar
Time
Tuesday, February 7, 2012 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Douglas A. LeonardAuburn University
Let I be an ideal in a polynomial ring R := F[x_n,...,x_1] Let A := R/I be the corresponding quotient ring, and let Q(A) be its field of fractions. The integral closure C(A, Q(A)) of A in Q(A) is a subring of the latter. But it is often given as a separate quotient ring, a presentation.Surprisingly, different computer algebra systems (Magma, Macaulay2, and Singular) choose to produce very different presentations. Some of these opt for presentations that have seductive forms, but miss the most important, namely a form that allows for determining when elements of Q(A) are in C(A,Q(A)). This is called membership and is directly related to determining isomorphism.

Using Mass formulas to Enumerate Definite Quadratic Forms of Bounded Class Number

Series
Algebra Seminar
Time
Tuesday, January 24, 2012 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jonathan HankeUniversity of Georgia
This talk will describe some recent results using exact massformulas to determine all definite quadratic forms of small class number inn>=3 variables, particularly those of class number one.The mass of a quadratic form connects the class number (i.e. number ofclasses in the genus) of a quadratic form with the volume of its adelicstabilizer, and is explicitly computable in terms of special values of zetafunctions. Comparing this with known results about the sizes ofautomorphism groups, one can make precise statements about the growth ofthe class number, and in principle determine those quadratic forms of smallclass number.We will describe some known results about masses and class numbers (overnumber fields), then present some new computational work over the rationalnumbers, and perhaps over some totally real number fields.

Pairs of polynomials over the rationals taking infinitely many common values

Series
Algebra Seminar
Time
Tuesday, January 10, 2012 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Benjamin WeissTechnion
For two polynomials G(X), H(Y) with rational coefficients, when does G(X) = H(Y) have infinitely many solutions over the rationals? Such G and H have been classified in various special cases by previous mathematicians. A theorem of Faltings (the Mordell conjecture) states that we need only analyze curves with genus at most 1.In my thesis (and more recent work), I classify G(X) = H(Y) defining irreducible genus zero curves. In this talk I'll present the infinite families which arise in this classification, and discuss the techniques used to complete the classification.I will also discuss in some detail the examples of polynomial which occur in the classification. The most interesting infinite family of polynomials are those H(Y) solving a Pell Equation H(Y)^2 - P(Y)Q(Y)^2 = 1. It turns out to be difficult to describe these polynomials more explicitly, and yet we can completely analyze their decompositions, how many such polynomials there are of a fixed degree, which of them are defined over the rationals (as opposed to a larger field), and other properties.

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